So thinking about it, my favorite psionic character is actually fairly weak- Gil the Arm, from Larry Niven's Known Space stories. Gil was an asteroid miner who lost his arm in an accident. But he developed phantom limb syndrome, where he was sure his arm was still there- and then, to his surprise, found he has a telekinetic limb, with the same reach and fine motor control as his old arm and hand, but very weak. After much practice, he can eventually lift a pen.
He returned to Earth and got a cloned arm, but it didn't affect his wild talent, and he eventually went to work for ARM (a sort of future FBI/Homeland Security organization). For having such a weak ability, he gets a lot done with it.
I really like the idea of psionic wild talents, though they were anything but balanced in 1e/2e, and they were really weak to the point of not being worth discussing in 4e's Dark Sun. I know there are Feats now that can grant psychic abilities, but it might be interesting to expand on this notion.
I don't like that they are Feats though. You get so few of them, and I get the feeling a lot of players wouldn't take them if they weren't free (which at least one probably would be in a Dark Sun campaign setting). But at the same time, there's not a heck of a lot you can give up in 5e for an alternate ability.
I wonder how trading Hit Dice for power would work? I'm sure my play experience is unique, but I've rarely had to spend Hit Dice to regain hit points, so I think that might be a worthwhile tradeoff, but I'm not sure.
That telekinetic limb sounds exactly like the Mage Hand cantrip.
I think it is ok if a background grants a standard cantrip.
When it comes to game design prioritize both balance and flavor.
A normal background grants about 3 tool/skill proficiencies plus what I call a "special noncombat asset". The exact number depends on how good the asset is.
The player and the DM together are supposed to create a new background if helpful for the character concept. Swapping proficiencies is routine.
A background is a small but useful amount design space.
For the DM, I feel the most important goal of a background is to encourage the player to feel connected to the imaginary world, and to continue investing in it in the future. The noncombat asset must be excellent. As long as the DM feels the player is enthusiastic about the STORY of the background, and its relationship to the persons and places in the setting, there is alot of things that a background can do.
In my opinion, a new language can be given for free, if, it is inherent to the background story.
On the safe side, suppose a non-asset design space of three decent skills or tools. Some skills are better than others. So where Animal Handling is usually worth about a point (but see below), Perception is about 2 points, and a decent skill, say like Stealth, is worth about 1-1/2 points. All three together are about 4-1/2 points. In other words, a normal background is worth about a feat, except that about half of it must be noncombat and narrative, rather than mechanical. Even so, a background still has about a strong half-feat of design space with mechanical teeth.
Remember, my variant backgrounds below are only if the DM feels the noncombat asset is sufficiently doing the heavy lifting for the setting storytelling.
The DM has 4 points to spend, and it is ok if it squeezes over to 4-1/2 points or falls slightly short at 3-1/2 points. When it comes to a background, the story is the priority.
1-1/2 points: decent skill or tool
1 point: one martial weapon
(This weapon must be central to the background, like a longbow for a deerhunter, or a longsword for a High Elf Military academy.)
1-1/2 points: all simple weapons
(a common medieval town militia)
2 points: shield
2 points: light armor
3 points: light+medium armor
(heavy armor including light and moderate is about 6 points and beyond the scope of a half-feat, but if a character already has medium from a class, the upgrade to heavy would be 3 points, but only if this really important to the background story.)
And now to the interesting stuff about
wild psionic backgrounds. Some cantrips are (way) better than other cantrips. A slot 1 spell once per day is within the scope of a background.
2-1/2 points: Guidance, Eldritch Blast, Minor Illusion, Mind Sliver, or Vicious Mockery
(Eldritch Blast deals force damage and can flavor appropriately as telekinetic choking, phantom gut punches, inducing heart attacks and similar diffuse trauma)
2 points: Mage Hand, Resistance, or Prestidigitation
1-1/2 points: Dancing Lights or Mending
1 point: Friends or Message
In 5e, skills are entirely the discretion of the DM. In my DM style, skills are super-important. It is a storytelling approach where the players narrate first person how they are interacting with the scene. If they are telling a story that seems to require some effort, there will probably be an ability check to doublecheck if I need to inject any information into the players story. You can see how valuable a skill bonus is compared to other mechanics.
I use skills in the following way:
Arcana = arcane magic only
Religion = divine magic only
Insight = psionic magic only
Nature = primal as element-and-plant
Animal Handling = beast magic
When used magically, and with their respective limitation, all skills are decent worth about 1-1/2 points each. Notice alchemy and elemental spells like Fireball require a Nature check to understand well. It might be Intelligence (Nature) depending on what the check is doublechecking.
So Gasik, especially because you are enthusiastic about the Gil the Arm character concept, as DM I would be open to negotiating a "psionic background" for it. I would require Insight for the sense of being intuitive about psionics.
PSIONIC BACKGROUND
Telekinetic Arm (Gil the Arm)
Mage Hand cantrip (2 points)
Insight (1-1/2 points)
Noncombat Asset: work for an investigatory security service.
AND I would allow 2 class skills to swap for the Investigation and Sleight Of Hand skills if not on the class list.
The point of a backgrouund is to tell the story. The DM should focus on a single variant, and narrarate with it as the centerpiece of a character concept. It is even better if the variant is noncombat, like the Mage Hand can be if the player is into it.
